#7003549

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,25,17,22)(16,28,30,21)(18,34,32,24)(20,33,36,27)(26,40,29,38)(31,39,41,43)(35,42,37,44);(5,6)(7,11)(8,10)(9,12)(13,15,18,27,19,17,32,33)(14,20,24,25,23,36,34,22)(16,29,31,44,30,26,41,42)(21,37,39,40,28,35,43,38);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,30)(14,21,23,28)(15,26,17,29)(18,31,32,41)(20,35,36,37)(22,40,25,38)(24,39,34,43)(27,42,33,44);(7,12)(9,11)(13,17,19,15)(14,22,23,25)(16,26,30,29)(18,27,32,33)(20,24,36,34)(21,38,28,40)(31,44,41,42)(35,43,37,39);(6,10)(9,11)(14,23)(16,30)(20,36)(22,25)(24,34)(26,29)(31,41)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910745);